NASA-TN-D-6283-1971 A time-dependent method for calculating supersonic angle-of-attack flow about axisymmetric blunt bodies with sharp shoulders and smooth nonaxisymmetric blunt bo.pdf

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1、NASA TECHNICAL NOTE A TIME-DEPENDENT METHOD FOR CALCULATING SUPERSONIC ANGLE-OF-ATTACK FLOW ABOUT AXISYMMETRIC BLUNT BODIES WITH SHARP SHOULDERS AND SMOOTH NONAXISYMMETRIC BLUNT BODIES by Richard W. Barnwell Langley Research Center Humpton, Vu. 23365 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WAS

2、HINGTON, D. C. AUGUST 1971 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM 1. Report No. I 2. Government Accession No. I 3. Recipients Catalog No. I NASA TN D-6283 4. Title and Subtitle A TIME-DEPENDENT METHOD FOR CALCU- LATING

3、SUPERSONIC ANGLE-OF-ATTACK FLOW ABOUT AXISYMMETRIC BLUNT BODIES WITH SHARP SHOULDERS AND SMOOTH NONAXISYMMETRIC BLUNT BODIES 7. Author(s1 Richard W. Barnwell 9. Performing Organization Name and Address NASA Langley Research Center Hampton, Va. 23365 12. Sponsoring Agency Name and Address National Ae

4、ronautics and Space Administration Washington, D.C. 20546 15. Supplementary Notes 5. Report Date August 1971 6. Performing Organization Code 8. Performing Organization Report No. L-7593 10. Work Unit No. 136-13-05-01 11. Contract or Grant No. 13. Type of Report and Period Covered Technical Note 14.

5、Sponsoring Agency Code 16. Abstract A time-dependent numerical method for calculating supersonic flow about blunt bodies at large angles of attack is presented. The axisymmetric bodies with sharp shoulders which are treated are constructed with a generator composed of segments of constant curvature.

6、 The nonaxisymmetric bodies have continuous slope and curvature. All flow fields are inviscid and adiabatic and have one plane of symmetry. A modification to the method of characteristics is introduced for use at the shock wave. A two-step finite-difference method of second-order accuracy is used at

7、 the body surface and in the region between the shock and body. A new finite-difference technique is introduced for use at sharp sonic shoulders. Comparisons of the results of the present method with experiment and the results of other methods are made for the flow of equilibrium air past the Apollo

8、 command module at the trim angle of attack and for perfect gas flow past a spherical cap and a spherically blunted cone at angle of attack. Both the cap and the blunted cone are terminated with sharp shoulders. Results are presented also for perfect gas flow past a prolate spheroid with its major a

9、xis normal to the flow. 17. Key Words (Suggested by Author(s) ) 1 18. Distribution Statement Angle of attack Blunt-body flow fields Sharp shoulders Unclassified - Unlimited Nonaxisymmetric bodies Supersonic, hypersonic Time-dependent finite-difference method 19. Security Classif. (of this report) -.

10、 - - I 20. Security Classif. (of this page) 22. Price. 21. No. of Pages Unclassified Unclassified $3.00 73 For sale by the National Technical Information Service, Springfield, Virginia 22151 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A TIME-DEPE

11、NDENT METHOD FOR CALCULATING SUPERSONIC ANGLE-OF-ATTACK FLOW ABOUT AXISYMMETRIC BLUNT BODIES WITH SHARP SHOULDERS AND SMOOTH NONAXISYMMETRIC BLUNT BODIES By Richard W. Barnwell Langley Research Center SUMMARY A time-dependent numerical method for calculating supersonic flow about blunt bodies at lar

12、ge angles of attack is presented. The axisymmetric bodies with sharp shoul- ders which are treated are constructed with a generator composed of segments of constant curvature. The nonaxisymmetric bodies have continuous slope and curvature. All flow fields are inviscid and adiabatic and have one plan

13、e of symmetry. A modification to the method of characteristics is introduced for use at the shock wave. A two-step finite-difference method of second-order accuracy is used at the body surface and in the region between the shock and body. A new finite-difference technique is introduced for use at sh

14、arp sonic shoulders. Comparisons of the results of the present method with experiment and the results of other methods are made for the flow of equilibrium air past the Apollo command module at the trim angle of attack and for perfect gas flow past a spherical cap and a spherically blunted cone at a

15、ngle of attack. Both the cap and the blunted cone are terminated with sharp shoulders. Results are presented also for perfect gas flow past a prolate spheroid with its major axis normal to the flow. INTRODUCTION Time-dependent finite-difference methods provide a means of treating the problem of invi

16、scid supersonic flow past a blunt body as an initial-value problem since the equa- tions for inviscid transient flow are always hyperbolic. Results for steady flow are obtained from the asymptotic solution to the transient problem. One of the major advan- tages of these methods is that there is no c

17、onceptual difficulty in extending them to treat such three-dimensional effects as angle of attack and nonaxisymmetric body geometry. In general, the chief difficulty encounteredin making a three-dimensional, rather than a Provided by IHSNot for ResaleNo reproduction or networking permitted without l

18、icense from IHS-,-,-two-dimensional, time-dependent calculation is the additional time required to perform the computation. Time-dependent methods for calculating three-dimensional blunt-body flow fields have been developed by Rusanov (ref. l), Bohachevsky and Mates (ref. 2), Moretti and Bleich (ref

19、. 3), and Xerikos and Anderson (ref. 4). Only the method of reference 1 can be applied to anything but axisymmetric bodies. All the methods except that of Moretti and Bleich produce results of first-order accuracy in the mesh spacings; the method of ref- erence 3 produces answers of second-order acc

20、uracy. The method of Bohachevsky and Mates requires a much larger number of grid points than the other methods because the bow shock wave is treated as an internal feature of the flow rather than as a boundary. As a result, a great deal more computer time is required for this method than for the oth

21、ers. Cohen, Foster, and Dowty (ref. 5) have employed the refined Godunov method of Masson, Taylor, and Foster (ref. 6) to develop an approximate time-dependent method for calculating angle-of-attack flow. The flow is calculated in the plane of symmetry, and trigonometric functions are used to approx

22、imate the cross-flow derivatives. The purpose of this paper is to present a time-dependent method for calculating three-dimensional flow fields about two fairly general classes of bodies. The first class is that of axisymmetric bodies and includes bodies with discontinuous surface slope and curvatur

23、e. The second class is that of nonaxisymmetric bodies with one plane of sym- metry and with continuous surface slope and curvature. Both perfect and equilibrium gas models can be treated. A previous version of this method for calculating flow about axi- symmetric bodies with sharp shoulders at angle

24、 of attack was presented in reference 7. The present method is a refinement of a previous method presented in references 8 and 9 and extends the applicability of that method to three dimensions. As was the case for the method of references 8 and 9, the present method is closely related to that of Mo

25、retti and coauthors (refs. 3 and 10). SYMBOLS D,E,E,F points in figures 4 and 5 i,B,C matrices defined by equations (B2) quantities defined by equation (6) 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-a speed of sound a71; semimajor and semimino

26、r axes of prolate spheroid “- Bi,Di,Ei quantities defined by equations (15) D - vector defined by equations (B2) E % matrix defined by equation (B11) e internal energy e eigenvalue of matrix E % u e quantity defined by equation (B17) - + eS unit vector normal to shock, defined by equation (26) - P u

27、nit vector in free-stream direction, defined by equation (31) v, gx,gy,gq unit vectors in x-, y-, and cp-directions, respectively F,Fc,Fo quantities used in equation (18) FL 7 Fu lower and upper bounds for inequality (B20) - FU(x,E) quantity defined by equation (B24) f quantity defined by equation (

28、B13) G quantity defined by equation (25) 6 matrix defined by equation (B12) g“ eigenvalue of matrix 5 H total enthalpy Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-h static enthalpy direction cosines of bicharacteristic in t,x,y,cp space K m m - p

29、1,p3 P PO Pt Q R r rn curvature of reference surface wavelength of error solutions in x-, y-, and cp-directions, respectively Mach number quantity defined by equations (B15) quantity defined by equations (B22) quantities defined by equations (B22) pressure reference pressure, 1 atmosphere (101.3 kN/

30、m2) exact value of stagnation pressure quantity defined by equation (B23) quantity equal to right side of equation (33) or (35) perpendicular distance from coordinate axis nose radius quantities defined by equations (B8) distance along surface from axis in plane of constant cp time components of vel

31、ocity tangent and normal to shock, respectively 4 . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-components of U in x-, y-, and q-directions, respectively quantity defined by equations (B15) components of velocity in x-, y-, and cp-directions, r

32、espectively velocity components shown in figure 5 magnitude of total velocity quantity defined by equation (B3) vector defined by equations (B2) distance along generator of reference surface from axis normalized coordinate defined by equations (7) coordinate along normals to reference surface coordi

33、nate along axis angle of attack exponent used in equation (B6) quantities defined by equations (30) quantities defined by equations (24) ratio of specific heats ratio of static enthalpy to mesh spacings for time internal energy, h/e T and coordinates X, Y, and 5 Provided by IHSNot for ResaleNo repro

34、duction or networking permitted without license from IHS-,-,-6 E shock layer thickness, function of T,X, damping coefficient r distance from corner, ,i - x ,v,t,* quantities defined by equations (B7) e angle between normal to reference surface and axis x scale factor for x-coordinate defined by equa

35、tion (3) pl,p2,p3 quantities defined by equations (B15) V exponent employed in equation (18) vl, v2, v3 quantities defined by equations (B22) P density c angle between normal to shock and free-stream direction 7 time +,Cp azimuthal angle X quantity which varies from 0 to 1 - w small nonnegative numb

36、er Subscripts : b properties at body C properties at shoulder I initial solution 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-max maximum value min minimum value S properties at shock 00 properties in free stream 900 properties for q = goo Super

37、scripts: W exponent between 0 and 1 * conditions when u = a and v = 0 differentiation with respect to t or T of functions which do not depend on Y differentiation with respect to x or X of functions which do not depend on Y differentiation with respect to cp or + of functions which do not depend on

38、Y vector quantity ANALYSIS The present method for calculating numerical solutions for time-dependent, inviscid, three-dimensional flow past blunt bodies traveling at supersonic speeds is described in this section. A number of grid points are located on the bow shock wave, the body sur- face, and bet

39、ween the shock and surface. The region of computation must contain the entire zone of subsonic flow. An initial solution, which can be quite general, is assumed; and the flow at each of the grid points is calculated for a number of time steps. At each cycle of the computation an initial-value proble

40、m is solved to determine the solution at the new time step from the solution at the previous time step, subject to the appropriate boundary conditions. Results for steady flow are obtained after many time steps when the time derivatives of the flow properties are sufficiently small. It should be not

41、ed that the locations of the grid points adjust with time as the location of the bow shock adjusts. 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A two-step finite-difference approximation to the time-dependent method of character- istics is used

42、 at the bow shock wave, whereas the two-step, time-dependent, finite- difference scheme of Brailovskaya (ref. 11) is used at the surface and between the shock and surface. The solution for a given time step is determined first at the points on the shock wave, then at those between the shock and body

43、, and finally at those on the body surface. Both the present method and the method of references 8 and 9 can be used to calcu- late axisymmetric flow about blunt bodies with sharp sonic shoulders. A major advantage of the present method is that it is not necessary to specify any of the flow properti

44、es at the shoulder. With the method of references 8 and 9, it is necessary to specify that the Mach number at the shoulder is 1 when the flow upstream of the shoulder is subsonic. A second advantage of the present method is that the computational techniques which are used at the bow shock wave and b

45、ody surface are much more efficient than the conven- tional time-dependent method of characteristics which is used in references 8 and 9. Basic Coordinate System and Governing Equations The basic coordinate system which is used is similar to that employed in refer- ences 8 and 9. An axisymmetric ref

46、erence surface is established as shown in figure 1. The coordinates are the azimuthal angle cp and the distances x and y, which are measured along the reference surface and normal to it, respectively, in planes of con- stant cp. The components of velocity in the x-, y-, and cp-directions are u, v, a

47、nd w. The angle in a plane of constant cp between the normal to the reference surface and the direction of the coordinate axis is designated as 8 and satisfies the differential equation where K is the local curvature of the reference surface. The distance r from the axis is given by the equation As

48、statea previously, the bodies which the present method will treat have one plane of symmetry. The free-stream velocity vector is parallel to this plane. The angle cp is measured from the leeward side of this plane, and the reference surface is constructed of segments of constant curvature. The refer

49、ence surface shown in figure 1 is constructed of one segment and hence is spherical. The type of coordinate system to be used with an axisymmetric body with a sharp shoulder is shown in figure 2. The reference surface which is used with the body in the 8 ,11111 I Provided by IHSNot for ResaleNo reproduction or networking